Deconstructing Complexity: A Computational Topology Approach to Trajectory Inference in the Human Thymus with tviblindi

Reviewer #1 (Public Review):

Summary:

The authors present tviblindi, a computational workflow for trajectory inference from molecular data at single-cell resolution. The method is based on (i) pseudo-time inference via expecting hitting time, (ii) sampling of random walks in a directed acyclic k-NN where edges are oriented away from a cell of origin w.r.t. the involved nodes' expected hitting times, and (iii) clustering of the random walks via persistent homology. An extended use case on mass cytometry data shows that tviblindi can be used elucidate the biology of T cell development.

Strengths:

- Overall, the paper is very well written and most (but not all, see below) steps of the tviblindi algorithm are explained well.

- The T cell biology use case is convincing (at least to me: I'm not an immunologist, only a bioinformatician with a strong interest in immunology).

Weaknesses:

- The main weakness of the paper is that a systematic comparison of tviblindi against other tools for trajectory inference (there are many) is entirely missing. Even though I really like the algorithmic approach underlying tviblindi, I would therefore not recommend to our wet-lab collaborators that they should use tviblindi to analyze their data. The only validation in the manuscript is the T cell development use case. Although this use case is convincing, it does not suffice for showing that the algorithms's results are systematically trustworthy and more meaningful (at least in some dimension) than trajectories inferred with one of the many existing methods.

- The authors' explanation of the random walk clustering via persistent homology in the Results (subsection "Real-time topological interactive clustering") is not detailed enough, essentially only concept dropping. What does "sparse regions" mean here and what does it mean that "persistent homology" is used? The authors should try to better describe this step such that the reader has a chance to get an intuition how the random walk clustering actually works. This is especially important because the selection of sparse regions is done interactively. Therefore, it's crucial that the users understand how this selection affects the results. For this, the authors must manage to provide a better intuition of the maths behind clustering of random walks via persistent homology.

- To motivate their work, the authors write in the introduction that "TI methods often use multiple steps of dimensionality reduction and/or clustering, inadvertently introducing bias. The choice of hyperparameters also fixes the a priori resolution in a way that is difficult to predict." They claim that tviblindi is better than the original methods because "analysis is performed in the original high-dimensional space, avoiding artifacts of dimensionality reduction." However, in the manuscript, tviblindi is tested only on mass cytometry data which has a much lower dimensionality than scRNA-seq data for which most existing trajectory inference methods are designed. Since tviblindi works on a k-NN graph representation of the input data, it is unclear if it could be run on scRNA-seq data without prior dimensionality reduction. For this, cell-cell distances would have to be computed in the original high-dimensional space, which is problematic due to the very high dimensionality of scRNA-seq data. Of course, the authors could explicitly reduce the scope of tviblindi to data of lower dimensionality, but this would have to be stated explicitly.

- Also tviblindi has at least one hyper-parameter, the number k used to construct the k-NN graphs (there are probably more hidden in the algorithm's subroutines). I did not find a systematic evaluation of the effect of this hyper-parameter.

Reviewer #2 (Public Review):

Summary: In Deconstructing Complexity: A Computational Topology Approach to Trajectory Inference in the Human Thymus with tviblindi, Stuchly et al. propose a new trajectory inference algorithm called tviblindi and a visualization algorithm called vaevictis for single-cell data. The paper utilizes novel and exciting ideas from computational topology coupled with random walk simulations to align single cells onto a continuum. The authors validate the utility of their approach largely using simulated data and establish known protein expression dynamics along CD4/CD8 T cell development in thymus using mass cytometry data. The authors also apply their method to track Treg development in single-cell RNA-sequencing data of human thymus.

The technical crux of the method is as follows: The authors provide an interactive tool to align single cells along a continuum axis. The method uses expected hitting time (given a user input start cell) to obtain a pseudotime alignment of cells. The pseudotime gives an orientation/direction for each cell, which is then used to simulate random walks. The random walks are then arranged/clustered based on the sparse region in the data they navigate using persistent homology.

Strengths:
The notion of using persistent homology to group random walks to identify trajectories in the data is novel.
The strength of the method lies in the implementation details that make computationally demanding ideas such as persistent homology more tractable for large scale single-cell data. This enables the authors to make the method more user friendly and interactive allowing real-time user query with the data.

Weaknesses:
The interactive nature of the tool is also a weakness, by allowing for user bias leading to possible overfitting for a specific data.

The main weakness of the method is lack of benchmarking the method on real data and comparison to other methods. Trajectory inference is a very crowded field with many highly successful and widely used algorithms, the two most relevant ones (closest to this manuscript) are not only not benchmarked against, but also not sited. Including those that specifically use persistent homology to discover trajectories (Rizvi et.al. published Nat Biotech 2017). Including those that specifically implement the idea of simulating random walks to identify stable states in single-cell data (e.g. CellRank published in Lange et.al Nat Meth 2022), as well as many trajectory algorithms that take alternative approaches. The paper has much less benchmarking, demonstration on real data and comparison to the very many other previous trajectory algorithms published before it. Generally speaking, in a crowded field of previously published trajectory methods, I do not think this one approach will compete well against prior work (especially due to its inability to handle the noise typical in real world data (as was even demonstrated in the little bit of application to real world data provided).

Beyond general lack of benchmarking there are two issues that give me particular concern. As previously mentioned, the algorithm is highly susceptible to user bias and overfitting. The paper gives the example (Figure 4) of a trajectory which mistakenly shows that cells may pass from an apoptotic phase to a different developmental stage. To circumvent this mistake, the authors propose the interactive version of tviblindi that allows users to zoom in (increase resolution) and identify that there are in fact two trajectories in one. In this case, the authors show how the author can fix a mistake when the answer is known. However, the point of trajectory inference is to discover the unknown. With so much interactive options for the user to guide the result, the method is more user/bias driven than data-driven. So a rigorous and quantitative discussion of robustness of the method, as well as how to ensure data-driven inference and avoid over-fitting would be useful.

Second, the paper discusses the benefit of tviblindi operating in the original high dimensions of the data. This is perhaps adequate for mass cytometry data where there is less of an issue of dropouts and the proteins may be chosen to be large independent. But in the context of single-cell RNA-sequencing data, the massive undersampling of mRNA, as well as high degree of noise (e.g. ambient RNA), introduces very large degree of noise so that modeling data in the original high dimensions leads to methods being fit to the noise. Therefore ALL other methods for trajectory inference work in a lower dimension, for very good reason, otherwise one is learning noise rather than signal. It would be great to have a discussion on the feasibility of the method as is for such noisy data and provide users with guidance. We note that the example scRNA-seq data included in the paper is denoised using imputation, which will likely result in the trajectory inference being oversmoothed as well.

Reviewer #3 (Public Review):

Summary:
Stuchly et al. proposed a single-cell trajectory inference tool, tviblindi, which was built on a sequential implementation of the k-nearest neighbor graph, random walk, persistent homology and clustering, and interactive visualization. The paper was organized around the detailed illustration of the usage and interpretation of results through the human thymus system.

Strengths:
Overall, I found the paper and method to be practical and needed in the field. Especially the in-depth, step-by-step demonstration of the application of tviblindi in numerous T cell development trajectories and how to interpret and validate the findings can be a template for many basic science and disease-related studies. The videos are also very helpful in showcasing how the tool works.

Weaknesses:
I only have a few minor suggestions that hopefully can make the paper easier to follow and the advantage of the method to be more convincing.
(1) The "Computational method for the TI and interrogation - tviblindi" subsection under the Results is a little hard to follow without having a thorough understanding of the tviblindi algorithm procedures. I would suggest that the authors discuss the uniqueness and advantages of the tool after the detailed introduction of the method (moving it after the "Connectome - a fully automated pipeline".
Also, considering it is a computational tool paper, inevitably, readers are curious about how it functions compared to other popular trajectory inference approaches. I did not find any formal discussion until almost the end of the supplementary note (even that is not cited anywhere in the main text). Authors may consider improving the summary of the advantages of tviblindi by incorporating concrete quantitative comparisons with other trajectory tools.
(2) Regarding the discussion in Figure 4 the trajectory goes through the apoptotic stage and reconnects back to the canonical trajectory with counterintuitive directionality, it can be a checkpoint as authors interpret using their expert knowledge, or maybe a false discovery of the tool. Maybe authors can consider running other algorithms on those cells and see which tracks they identify and if the directionality matches with the tviblindi.
(3) The paper mainly focused on mass cytometry data and had a brief discussion on scRNA-seq. Can the tool be applied to multimodality data such as CITE-seq data that have both protein markers and gene expression? Any suggestions if users want to adapt to scATAC-seq or other epigenomic data?

Author response:

eLife assessment

The authors present an algorithm and workflow for the inference of developmental trajectories from single-cell data, including a mathematical approach to increase computational efficiency. While such efforts are in principle useful, the absence of benchmarking against synthetic data and a wide range of different single-cell data sets make this study incomplete. Based on what is presented, one can neither ultimately judge if this will be an advance over previous work nor whether the approach will be of general applicability.

We thank the eLife editor for the valuable feedback. We wish to emphasize that both, benchmarking against other methods and validation on a synthetic dataset (“dyntoy”) are indeed presented in Supplementary Note, although we failed to sufficiently emphasize it in the main text.

We will extend the benchmarking to more TI methods and we will improve the results and discussion sections to present those facts more clearly to the reader.

Public Reviews:

Reviewer #1 (Public Review):

Summary:

The authors present tviblindi, a computational workflow for trajectory inference from molecular data at single-cell resolution. The method is based on (i) pseudo-time inference via expecting hitting time, (ii) sampling of random walks in a directed acyclic k-NN where edges are oriented away from a cell of origin w.r.t. the involved nodes' expected hitting times, and (iii) clustering of the random walks via persistent homology. An extended use case on mass cytometry data shows that tviblindi can be used elucidate the biology of T cell development.

Strengths:

- Overall, the paper is very well written and most (but not all, see below) steps of the tviblindi algorithm are explained well.

- The T cell biology use case is convincing (at least to me: I'm not an immunologist, only a bioinformatician with a strong interest in immunology).

We thank the reviewer for feedback and suggestions that we will accommodate, we respond point-by-point below

Weaknesses:

- The main weakness of the paper is that a systematic comparison of tviblindi against other tools for trajectory inference (there are many) is entirely missing. Even though I really like the algorithmic approach underlying tviblindi, I would therefore not recommend to our wet-lab collaborators that they should use tviblindi to analyze their data. The only validation in the manuscript is the T cell development use case. Although this use case is convincing, it does not suffice for showing that the algorithms's results are systematically trustworthy and more meaningful (at least in some dimension) than trajectories inferred with one of the many existing methods.

We have compared tviblindi to several trajectory inference methods (Supplementary note section 8.2: Comparison to state-of-the-art methods, namely Monocle3 (v1.3.1) Cao et al. (2019), Stream (v1.1) Chen et al. (2019), Palantir (v1.0.0) Setty et al. (2019), VIA (v0.1.89) Stassen et al. (2021) and PAGA (scanpy==1.9.3) Wolf et al. (2019).) We will add thorough and systematic comparisons to the other algorithms mentioned by reviewers. We will include extended evaluation on publically available datasets.

Also, we have successfully used tviblindi to investigate human B-cell development in primary immunodeficiency (manuscript in revisions), double negative T-cells development in ALPS (Autoimmune Lymphoproliferative Syndrome) by mass cytometry (project in progress).

- The authors' explanation of the random walk clustering via persistent homology in the Results (subsection "Real-time topological interactive clustering") is not detailed enough, essentially only concept dropping. What does "sparse regions" mean here and what does it mean that "persistent homology" is used? The authors should try to better describe this step such that the reader has a chance to get an intuition how the random walk clustering actually works. This is especially important because the selection of sparse regions is done interactively. Therefore, it's crucial that the users understand how this selection affects the results. For this, the authors must manage to provide a better intuition of the maths behind clustering of random walks via persistent homology.

In order to satisfy both reader types: the biologist and the mathematician, we explain the mathematics in detail in the Supplementary Note, section 4. We will improve the Results text to better point the reader to the mathematical foundations in the Supplementary Note.

- To motivate their work, the authors write in the introduction that "TI methods often use multiple steps of dimensionality reduction and/or clustering, inadvertently introducing bias. The choice of hyperparameters also fixes the a priori resolution in a way that is difficult to predict." They claim that tviblindi is better than the original methods because "analysis is performed in the original high-dimensional space, avoiding artifacts of dimensionality reduction." However, in the manuscript, tviblindi is tested only on mass cytometry data which has a much lower dimensionality than scRNA-seq data for which most existing trajectory inference methods are designed. Since tviblindi works on a k-NN graph representation of the input data, it is unclear if it could be run on scRNA-seq data without prior dimensionality reduction. For this, cell-cell distances would have to be computed in the original high-dimensional space, which is problematic due to the very high dimensionality of scRNA-seq data. Of course, the authors could explicitly reduce the scope of tviblindi to data of lower dimensionality, but this would have to be stated explicitly.

In the manuscript we tested the framework on the scRNA-seq data from Park et al 2020 (DOI: 10.1126/science.aay3224). To illustrate that tviblindi can work directly in the high-dimensional space, we applied the framework successfully on imputed 2000 dimensional data.

The idea behind tviblindi is to be able to work without the necessity to use non-linear dimensionality reduction techniques, which reduce the dimensionality to a very low number of dimensions and whose effects on the data distribution are difficult to predict. On the other hand the use of (linear) dimensionality reduction techniques which effectively suppress noise in the data such as PCA is a good practice (see also response to reviewer 2). We will emphasize this in the revised version and add the results of the corresponding analysis.

- Also tviblindi has at least one hyper-parameter, the number k used to construct the k-NN graphs (there are probably more hidden in the algorithm's subroutines). I did not find a systematic evaluation of the effect of this hyper-parameter.

Detailed discussion of the topic is presented in the Supplementary Note, section 8.1, where Spearman correlation coefficient between pseudotime estimated using k=10 and k=50 nearest neighbors was 0.997. The number k however does affect the number of candidate endpoints. But even when larger k causes spurious connection between unrelated cell fates, the topological clustering of random walks allows for the separation of different trajectories. We will expand the “sensitivity to hyperparameters section” also in response to reviewer 2.

Reviewer #2 (Public Review):

Summary:

In Deconstructing Complexity: A Computational Topology Approach to Trajectory Inference in the Human Thymus with tviblindi, Stuchly et al. propose a new trajectory inference algorithm called tviblindi and a visualization algorithm called vaevictis for single-cell data. The paper utilizes novel and exciting ideas from computational topology coupled with random walk simulations to align single cells onto a continuum. The authors validate the utility of their approach largely using simulated data and establish known protein expression dynamics along CD4/CD8 T cell development in thymus using mass cytometry data. The authors also apply their method to track Treg development in single-cell RNA-sequencing data of human thymus.

The technical crux of the method is as follows: The authors provide an interactive tool to align single cells along a continuum axis. The method uses expected hitting time (given a user input start cell) to obtain a pseudotime alignment of cells. The pseudotime gives an orientation/direction for each cell, which is then used to simulate random walks. The random walks are then arranged/clustered based on the sparse region in the data they navigate using persistent homology.

We thank the reviewer for feedback and suggestions that we will accommodate, we respond point-by-point below.

Strengths:

The notion of using persistent homology to group random walks to identify trajectories in the data is novel.

The strength of the method lies in the implementation details that make computationally demanding ideas such as persistent homology more tractable for large scale single-cell data.

This enables the authors to make the method more user friendly and interactive allowing real-time user query with the data.

Weaknesses:

The interactive nature of the tool is also a weakness, by allowing for user bias leading to possible overfitting for a specific data.

tviblindi is not designed as a fully automated TI tool (although it implements a fully automated module), but as a data driven framework for exploratory analysis of unknown data. There is always a risk of possible bias in this type of analysis - starting with experimental design, choice of hyperparameters in the downstream analysis, and an expert interpretation of the results. The successful analysis of new biological data involves a great deal of expert knowledge which is difficult to a priori include in the computational models.

tvilblindi tries to solve this challenge by intentionally overfitting the data and keeping the level of resolution on a single random walk. In this way we aim to capture all putative local relationships in the data. The on-demand aggregation of the walks using the global topology of the data allows researchers to use their expert knowledge to choose the right level of detail (as demonstrated in the Figure 4 of the manuscript) while relying on the topological structure of the high dimensional point cloud. At all times tviblindi allows to inspect the composition of the trajectory to assess the variance in the development, possible hubs on the KNN-graph etc.

The main weakness of the method is lack of benchmarking the method on real data and comparison to other methods. Trajectory inference is a very crowded field with many highly successful and widely used algorithms, the two most relevant ones (closest to this manuscript) are not only not benchmarked against, but also not sited. Including those that specifically use persistent homology to discover trajectories (Rizvi et.al. published Nat Biotech 2017). Including those that specifically implement the idea of simulating random walks to identify stable states in single-cell data (e.g. CellRank published in Lange et.al Nat Meth 2022), as well as many trajectory algorithms that take alternative approaches. The paper has much less benchmarking, demonstration on real data and comparison to the very many other previous trajectory algorithms published before it. Generally speaking, in a crowded field of previously published trajectory methods, I do not think this one approach will compete well against prior work (especially due to its inability to handle the noise typical in real world data (as was even demonstrated in the little bit of application to real world data provided).

We provide comparisons of tviblindi and vaevictis in the Supplementary Note, section 8.2, where we compare it to Monocle3 (v1.3.1) Cao et al. (2019), Stream (v1.1) Chen et al. (2019), Palantir (v1.0.0) Setty et al. (2019), VIA (v0.1.89) Stassen et al. (2021) and PAGA (scanpy==1.9.3) Wolf et al. (2019). We use two datasets: artificial Dyntoy and real mass cytometry thymus+peripheral blood dataset. We thank the reviewer for suggesting specific methods. CellRank was excluded from the benchmarking as it was originally designed for RNA-velocity data (not available in mass cytometry data), but will include recent upgrade CellRank2 (preprint at doi.org/10.1101/2023.07.19.549685) which offers more flexibility.

We will add further benchmarking as suggested by the reviewer in the course of revisions.

Beyond general lack of benchmarking there are two issues that give me particular concern. As previously mentioned, the algorithm is highly susceptible to user bias and overfitting. The paper gives the example (Figure 4) of a trajectory which mistakenly shows that cells may pass from an apoptotic phase to a different developmental stage. To circumvent this mistake, the authors propose the interactive version of tviblindi that allows users to zoom in (increase resolution) and identify that there are in fact two trajectories in one. In this case, the authors show how the author can fix a mistake when the answer is known. However, the point of trajectory inference is to discover the unknown. With so much interactive options for the user to guide the result, the method is more user/bias driven than data-driven. So a rigorous and quantitative discussion of robustness of the method, as well as how to ensure data-driven inference and avoid over-fitting would be useful.

Local directionality in expression data is a challenge which is not, to our knowledge, solved. And we are not sure it can be solved entirely, even theoretically. The random walks passing “through” the apoptotic phase are biologically infeasible, but it is an (unbiased) representation of what the data look like based on the diffusion model. It is a property of the data (or of the panel design), which has to be interpreted properly rather than a mistake. Of note, except for Monocle3 (which does not provide the directionality) other tested methods did not discover this trajectory at all.

The “zoom in” has in fact nothing to do with “passing through the apoptosis”. We show how the researcher can investigate the suggested trajectory to see if there is an additional structure of interest and/or relevance. This investigation is still data driven (although not fully automated). Anecdotally in this particular case this branching was discovered by an bioinformatician, who knew nothing about the presence of beta-selection in the data.

We show that the trajectory of apoptosis of cortical thymocytes consists of 2 trajectories corresponding to 2 different checkpoints (beta-selection and positive/negative selection). This type of structure, where 2 (or more) trajectories share the same path for most of the time, then diverge only to be connected at a later moment (immediately from the point of view of the beta-selection failure trajectory) is a challenge for TI algorithms and none of tested methods gave a correct result. More importantly there seems to be no clear way to focus on these kinds of structures (common origin and common fate) in TI methods.

Of note, the “zoom in” is a recommended and convenient method to look for an inner structure, but it does not necessarily mean addition of further homological classes. Indeed, in this case the reason that the structure is not visible directly is the limitation of the dendrogram complexity (only branches containing at least 10% of simulated random walks are shown by default).

In summary, tviblindi effectively handled all noise in the data that obscured biologically valid trajectories for other methods. We will improve the discussion of the robustness in the reviewed version.

Second, the paper discusses the benefit of tviblindi operating in the original high dimensions of the data. This is perhaps adequate for mass cytometry data where there is less of an issue of dropouts and the proteins may be chosen to be large independent. But in the context of single-cell RNA-sequencing data, the massive undersampling of mRNA, as well as high degree of noise (e.g. ambient RNA), introduces very large degree of noise so that modeling data in the original high dimensions leads to methods being fit to the noise. Therefore ALL other methods for trajectory inference work in a lower dimension, for very good reason, otherwise one is learning noise rather than signal. It would be great to have a discussion on the feasibility of the method as is for such noisy data and provide users with guidance. We note that the example scRNA-seq data included in the paper is denoised using imputation, which will likely result in the trajectory inference being oversmoothed as well.

We agree with the reviewer. In our manuscript we wanted to showcase that tviblindi can directly operate in high-dimensional space (thousands of dimensions) and we used MAGIC imputation for this purpose. This was not ideal. More standard approach, which uses 30-50 PCs as input to the algorithm resulted in equivalent trajectories. We will add this analysis to the study.

In summary, the fact that tviblindi scales well with dimensionality of the data and is able to work in the original space does not mean that it is always the best option. We will emphasize in the revised paper that we aim to avoid the non-linear dimensional reduction techniques as a data preprocessing tool, as the effect of the reduction is difficult to predict. We will also discuss the preprocessing of scRNA-seq data in greater detail.

Reviewer #3 (Public Review):

Summary:

Stuchly et al. proposed a single-cell trajectory inference tool, tviblindi, which was built on a sequential implementation of the k-nearest neighbor graph, random walk, persistent homology and clustering, and interactive visualization. The paper was organized around the detailed illustration of the usage and interpretation of results through the human thymus system.

Strengths:

Overall, I found the paper and method to be practical and needed in the field. Especially the in-depth, step-by-step demonstration of the application of tviblindi in numerous T cell development trajectories and how to interpret and validate the findings can be a template for many basic science and disease-related studies. The videos are also very helpful in showcasing how the tool works.

Weaknesses:

I only have a few minor suggestions that hopefully can make the paper easier to follow and the advantage of the method to be more convincing.

(1) The "Computational method for the TI and interrogation - tviblindi" subsection under the Results is a little hard to follow without having a thorough understanding of the tviblindi algorithm procedures. I would suggest that the authors discuss the uniqueness and advantages of the tool after the detailed introduction of the method (moving it after the "Connectome - a fully automated pipeline".

We thank the reviewer for the suggestion and we will accommodate it to improve readability of the text.

Also, considering it is a computational tool paper, inevitably, readers are curious about how it functions compared to other popular trajectory inference approaches. I did not find any formal discussion until almost the end of the supplementary note (even that is not cited anywhere in the main text). Authors may consider improving the summary of the advantages of tviblindi by incorporating concrete quantitative comparisons with other trajectory tools.

We provide comparisons of tviblindi and vaevictis in the Supplementary Note, section 8.2, where we compare it to Monocle3 (v1.3.1) Cao et al. (2019), Stream (v1.1) Chen et al. (2019), Palantir (v1.0.0) Setty et al. (2019), VIA (v0.1.89) Stassen et al. (2021) and PAGA (scanpy==1.9.3) Wolf et al. (2019). We use two datasets: artificial Dyntoy and real mass cytometry thymus+peripheral blood dataset. We will also add CellRank2 into comparisons and we will strengthen the message of the benchmarking results in the Discussion section.

(2) Regarding the discussion in Figure 4 the trajectory goes through the apoptotic stage and reconnects back to the canonical trajectory with counterintuitive directionality, it can be a checkpoint as authors interpret using their expert knowledge, or maybe a false discovery of the tool. Maybe authors can consider running other algorithms on those cells and see which tracks they identify and if the directionality matches with the tviblindi.

We have indeed used the thymus dataset for comparison of all TI algorithms listed above. Except for Monocle 3 they failed to discover the negative selection branch (Monocle 3 does not offer directionality information). Therefore, a valid topological trajectory with incorrect (expert-corrected) directionality was partly or entirely missed by other algorithms.

(3) The paper mainly focused on mass cytometry data and had a brief discussion on scRNA-seq. Can the tool be applied to multimodality data such as CITE-seq data that have both protein markers and gene expression? Any suggestions if users want to adapt to scATAC-seq or other epigenomic data?

The analysis of multimodal data is the logical next step and is the topic of our current research. At this moment tviblindi cannot be applied directly to multimodal data. It is possible to use the KNN-graph based on multimodal data (such as weighted nearest neighbor graph implemented in Seurat) for pseudotime calculation and random walk simulation. However, we do not have a fully developed triangulation for the multimodal case yet.